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Prescription of finite Dirichlet eigenvalues and area on surface with boundary

Xiang He

Abstract

In the present paper, we consider Dirichlet Laplacian on compact surface. We show that for a fixed surface with boundary $X$, a finite increasing sequence of real numbers $0<a_1<a_2<\cdots<a_N$ and a positive number $A$, there exists a metric $g$ on $X$ such that for any integer $1\leq k\leq N$, we have $λ_k^\mathcal{D}(X,g)=a_k$ and $\mathrm{Area}(X,g)=A$.

Prescription of finite Dirichlet eigenvalues and area on surface with boundary

Abstract

In the present paper, we consider Dirichlet Laplacian on compact surface. We show that for a fixed surface with boundary , a finite increasing sequence of real numbers and a positive number , there exists a metric on such that for any integer , we have and .
Paper Structure (7 sections, 12 theorems, 70 equations)

This paper contains 7 sections, 12 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. The proof of Theorem \ref{['mthm']}
  3. Eigenvalues on the star graph with one point boundary
  4. Reduce the problem to the case of disk
  5. Proof of the first part of Theorem \ref{['mthm']} for the disk
  6. Proof of the second part of Theorem \ref{['mthm']}
  7. Appendix

Key Result

Theorem 1.1

Let $X$ be a smooth surface with nonempty boundary, $0<a_1<a_2<\cdots<a_N$ be a given sequence of real numbers and $A$ be a given positive number. Then there exists a Riemannian metric $g$ on $X$ such that for any integer $1\leq k\leq N$, we have $\lambda_k^\mathcal{D}(X,g)=a_k$ and $\mathop{\mathrm

Theorems & Definitions (20)

  • Theorem 1.1
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6: Lemma 3.2 in JTBD
  • ...and 10 more